K-moduli of curves on a quadric surface and K3 surfaces

Abstract

We show that the K-moduli spaces of log Fano pairs $({\mathbb{P}}^1\times {\mathbb{P}}^1,cC)$ where $C$ is a $(4,4)$-curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ complete intersection curves in ${\mathbb{P}}^3$. This, together with recent results by Laza-O'Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$-curves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

34 pages. Comments are very welcome. v2: to appear in J. Inst. Math. Jussieu

References in corpus (5)

• CM Stability and the Generalized Futaki Invariant I
• K-stability of log Fano hyperplane arrangements
• Wall crossing for K-moduli spaces of plane curves
• Degenerations of del Pezzo surfaces I
• K-stability of cubic fourfolds

Cited by in corpus (3)

• K-stability of Fano varieties: an algebro-geometric approach
• K-stability of cubic fourfolds
• Polystable log Calabi-Yau varieties and Gravitational instantons